Diameter and stationary distribution of random r-out digraphs

Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The constant ¿r is related to branching processes and also appears in other model...

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Detalles Bibliográficos
Autores: Addario-Berry, Louigi, Balle, Borja, Perarnau Llobet, Guillem|||0000-0002-1953-9511
Tipo de recurso: artículo
Fecha de publicación:2020
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/342134
Acceso en línea:https://hdl.handle.net/2117/342134
https://dx.doi.org/10.37236/9485
Access Level:acceso abierto
Palabra clave:Graph theory
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
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spelling Diameter and stationary distribution of random r-out digraphsAddario-Berry, LouigiBalle, BorjaPerarnau Llobet, Guillem|||0000-0002-1953-9511Graph theoryGrafs, Teoria deÀrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafsLet D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The constant ¿r is related to branching processes and also appears in other models of random undirected graphs. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n, r). In particular, we determine the asymptotic behaviour of pmax and pmin, the maximum and the minimum values of the stationary distribution. We show that with high probability pmax = n -1+o(1) and pmin = n -(1+¿r)+o(1). Our proof shows that the vertices with p(v) near to pmin lie at the top of “narrow, slippery towers”; such vertices are also responsible for increasing the diameter from (1+o(1)) logr n to (1 + ¿r + o(1)) logr n.20202020-08-0720212021-03-22journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/342134https://dx.doi.org/10.37236/9485reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3421342026-05-27T15:37:01Z
dc.title.none.fl_str_mv Diameter and stationary distribution of random r-out digraphs
title Diameter and stationary distribution of random r-out digraphs
spellingShingle Diameter and stationary distribution of random r-out digraphs
Addario-Berry, Louigi
Graph theory
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
title_short Diameter and stationary distribution of random r-out digraphs
title_full Diameter and stationary distribution of random r-out digraphs
title_fullStr Diameter and stationary distribution of random r-out digraphs
title_full_unstemmed Diameter and stationary distribution of random r-out digraphs
title_sort Diameter and stationary distribution of random r-out digraphs
dc.creator.none.fl_str_mv Addario-Berry, Louigi
Balle, Borja
Perarnau Llobet, Guillem|||0000-0002-1953-9511
author Addario-Berry, Louigi
author_facet Addario-Berry, Louigi
Balle, Borja
Perarnau Llobet, Guillem|||0000-0002-1953-9511
author_role author
author2 Balle, Borja
Perarnau Llobet, Guillem|||0000-0002-1953-9511
author2_role author
author
dc.subject.none.fl_str_mv Graph theory
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
topic Graph theory
Grafs, Teoria de
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
description Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The constant ¿r is related to branching processes and also appears in other models of random undirected graphs. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n, r). In particular, we determine the asymptotic behaviour of pmax and pmin, the maximum and the minimum values of the stationary distribution. We show that with high probability pmax = n -1+o(1) and pmin = n -(1+¿r)+o(1). Our proof shows that the vertices with p(v) near to pmin lie at the top of “narrow, slippery towers”; such vertices are also responsible for increasing the diameter from (1+o(1)) logr n to (1 + ¿r + o(1)) logr n.
publishDate 2020
dc.date.none.fl_str_mv 2020
2020-08-07
2021
2021-03-22
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/342134
https://dx.doi.org/10.37236/9485
url https://hdl.handle.net/2117/342134
https://dx.doi.org/10.37236/9485
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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